A minimal model for excitable and bursting elements

*Corresponding author. Fax: #55-16633-9949.E-mail address: osame@dfm.!clrp.usp.br (O. Kinouchi).

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Neurocomputing 38}40 (2001) 255}261

A minimal model for excitable and bursting elements

Silvia M. Kuva�, Gilson F. Lima��, Osame Kinouchi��*,Marcelo H.R. Tragtenberg��, Anto( nio C. Roque�

�Laborato& rio de NeurocieL ncia Computational, Departamento de Fn&sica e Matema& tica, Faculdade de Filosoxa,CieL ncias e Letras de RibeiraJ o Preto, Universidade de SaJ o Paulo Av. dos Bandeirantes 3900, CEP 14040-901,

RibeiraJ o Preto, SP, Brazil�Condensed Matter Group, Department of Theoretical Physics, Oxford University, 1 Keble Road,

Oxford, UK OX1 3NP, UK�Departamento de Fn&sica*Universidade Federal de Santa Catarina, Trindade, Floriano& polis, SC, CEP 88040-

900, Brazil�Escola Te&cnica Federal de Mato Grosso, Cuiaba& , Mato Grosso, Brazil

Abstract

We propose a simple map (a dynamical system with discrete time) as a minimal formal modelof excitable and bursting cells. The map has two fast variables and a single slow one andpresents all the usual behavior of excitable cells like fast spiking, regular spiking, bursting,plateau action potentials and adaptation phenomena. The simplicity of the map enables us toexamine large regions in parameter space. The map can be used as a versatile element for largescale simulations of neural systems.We discuss the e$cient representation of chemical synapsesin these coupled maps lattices. � 2001 Elsevier Science B.V. All rights reserved.

Keywords: Excitable cells; Bursting cells; Neuron models; FitzHugh}Nagumo; Rose}Hin-dmarsh; Coupled maps lattices

1. Introduction

Coupled maps lattices (CML) have been successfully used in physics for studyinggeneric features of complex dynamics like pattern formation, spatio-temporal chaosand the onset of turbulence. Maps are dynamical systems with continuous state

0925-2312/01/$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 5 - 2 3 1 2 ( 0 1 ) 0 0 3 7 6 - 9

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variables but discrete-time dynamics. The use of CMLs in biological modeling hasbeen less explored [1,4,2], but they could be useful for the study of robust collectiveproperties of excitable cells.The proper use of CMLs in neural network modeling depends on the choice of

a good basic element. In the past some CML researchers have represented neurons bysimple elements like the one-dimensional logistic map [4], which seems not to bea good option to represent excitable cells. Two variable maps have been proposed byAihara et al. [1], Chialvo [2] and Kinouchi and Tragtenberg [6]. Further method-ological considerations about the use of maps in neural modeling may be found inthese papers.Here we study a three variable map (two fast and one slow) which is the minimal

number of variables for representing bursting cells. The map is a generalization of thatused in [6] and can be thought as a discrete-time counterpart of standard bursting cellmodels like the Rose}Hindmarsh system [3,5]. The form of the map is very simple butits dynamical behaviors (excitable "xed point, fast spiking, slow regular spiking, burstsand spikes with plateau) seems to be adequate to represent neurons and otherexcitable cells.

2. The bursting cell model

We propose the following three-dimensional non-linear map as our bursting cellmodel:

x(t#1)"tanh�x(t)!Ky(t)#z(t)#I(t)

¹ �,y(t#1)"x(t),

z(t#1)"(1!�)z(t)!�(x(t)!x�). (1)

Like in the Rose}Hindmarsh [3] model, the variable x represents the membranepotential, y is a recovery variable and z is a slow adaptive current (since we assumethat �,�;1). The map has "ve parameters: K, ¹, �, � and x

�. The external current

is I(t).Notice that the map can also be viewed as a formal sigmoid neuron with two

self-couplings

x(t#1)"tanh[�(w�x(t)#w

�x(t!1)#z(t)#I(t))] (2)

with gain �"1/¹, weights w�"1, w

�"!K and an adaptive bias z(t). Here,

however, x(t) represents the membrane potential (to be scaled by a dimensional factor)and not the "ring rate as in usual sigmoid neurons.The case �"�"0, that is, the two-dimensional case where z(t)"z is constant, has

been studied in Kinouchi and Tragtenberg [6]. These authors determined the para-meter region of interest as being the excitable regime near a sub-critical Hopfbifurcation (see Fig. 1).

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Fig. 1 . Phase diagram for the two-dimensional model for K"0.6. There are a single "xed point in the FPregion, and two "xed points (low and high x) in the 2FP region. In regionOsc there are cycles or oscillations(fast repetitive "ring). The stability line of the "xed point is given by the solid curve but the cycles only loosetheir stability at the dashed line, so there is a region of bistability between "xed points and cycles(FP#Osc).

3. Phase diagram

In Fig. 2 we present the phase diagram in the plane x�versus ¹, with "xed values

K"0.6, �"�"0.001. Similar plots can be made for other parameter values. Fivemain regimes are found: a "xed point region (FP), a region of fast repetitive spikes(FS), a region with spikes with plateau (cardiac-like C), an intermediary region withrepetitive bursts (B) and a region with sub-threshold oscillations (ST).When the slow current z(t) is present, it performs a slow oscillation along the

bistable regions on the z axis (vertical direction in Fig. 1). Basically, the z(t) variablegrows if the time averaged membrane potential is low and decreases when thepotential is high when compared to a reversal potential x

�.

If this modulation occurs over the bistable region where a "xed point (low restingstate) and a cycle coexist, then we have a type-III bursting mechanism [5,7] whererepetitive transitions between these two states occur. Notice that this occurs over thelower FP#Osc region in Fig. 1 (negative z).We also observe that for higher values of �, the number of spikes in a burst

decreases. We observe triplets, doublets and "nally singlets (say, for �"�"0.005).This singlet state could be identi"ed as a regular (slow) spiking regime. We havechecked that chaotic behavior (positive Lyapunov exponents) occurs mainly in thebursting region.If the modulation occurs over the two "xed point bistable region (2FP in Fig. 1),

then we have spikes with plateau (cardiac-like spikes). For higher ¹ or x�there is an

S.M. Kuva et al. / Neurocomputing 38}40 (2001) 255}261 257

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Fig. 2. Phase diagram in the parameter plane x�versus ¹ for K"0.6, �"�"0.001. The typical

behaviors are: "xed point (FP) cardiac-like spikes (C), repetitive bursts (B), fast spiking (FS), sub-thresholdoscillations (ST).

Fig. 3. Examples of typical behaviors, with parametersK"0.6, �"�"0.001 for (a)}(c) and �"�"0.005for (d): (a) Spikes with plateau, region C (¹"0.25, x

�"!0.5). (b) Bursts, region B (¹"0.35,x

�"!0.5).

(c) Fast spiking (¹"0.4,x�"!0.1). (d) Regular (slow) spiking (¹"0.35,x

�"!0.5).

equilibrium situation where z(t) is almost stationary, and the cell presents a repetitive"ring behavior.These di!erent situations are illustrated in Fig. 3. The width of a spike de"nes

a natural time scale, say 1 ms. A typical spike of our neuron model has width of tentime steps, so we have used the scale one time step "0.1 ms in Figs. 3 and 4.Notice that all these behaviors refer to repetitive spontaneous activity. If we desire

to model a quiescent excitable neuron, we must choose x�in the FP region, at the


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Fig. 4. Post-synaptic current time course due to a pre-synaptic burst. Notice the temporal summatione!ect. Activity x(t) in the pre-synaptic neuron (squares) and post-synaptic current f(t) in arbitrary units.Parameters: J"0.01, �

�"�

�"15 (synaptic map) and K"0.6, ¹"0.35, �"�"0.001, x

�"!0.5

(pre-synaptic neuron).

border of the transition to repetitive activity. When a external input I is applied, wealso observe adaptation phenomena: the element start "ring at a high frequency butthen the slow z current decreases and the activity diminishes or even stops, dependingon the � and � values.

4. Modeling chemical synapses

Instead of the usual di!usive coupling studied in most coupled maps latticeliterature, the coupling among neurons is accomplished by complicated structuressuch as chemical synapses. Now we describe how to introduce such synapses keepingthe discrete time nature of the network.A network ofN identical cells can be modeled by i"1,2,N tri-dimensional maps

x�(t#1)"tanh�

x�(t)!Ky

�(t)#z

�(t)#I

�(t)

¹ �,y�(t#1)"x

�(t),

z�(t#1)"(1!�)z

�(t)!�(x

�(t)!x

�), (3)

where I�(t)"�

�f��(t) is the total input to the ith neuron and f

��are post-synaptic

currents (PSC) produced by all the pre-synaptic neurons j coupled to cell i.


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We propose to represent each chemical synapse by a two variable system, which wewill call the synaptic map

f��(t#1)"�1!

1

�� f

��(t)#g

��(t),

g��(t#1)"�1!

1

��g�� (t)#J

��(x(t)).

(4)

In this bi-dimensional map, f represents the PSC's time course and g is an auxiliaryvariable. The synaptic strength J

��scales the PSC amplitude and the step function

(�(x)"1 for x'0, zero otherwise) detects the occurrence of a spike in the pre-synaptic neuron.It is possible to show that if �

�,��are large compared to the time step, then the

above map approximates well some standard functions used to model PSCs. If thetime constants are equal (�

�"�

�"�) the solution f (t) is very similar to the so called

alpha function, �(t)"Ct exp(!t/�). If ��O�

�, the solution f (t) approximates very

well the double exponential �(t)"C[exp(!t/��)!exp(!t/�

�)]. Since the charac-

teristic times of PSCs are always larger than the spike width (which corresponds tonear ten time steps in our cell map), the above approximation condition is alwaysvalid.This synaptic representation automatically incorporates temporal summation

(Fig. 4). But its principal virtue is that both neurons and synaptic currents are updatedin the same form, at the same time steps, without the need of keeping a list of "ringtimes or other arti"cial coupling schemes.

5. Conclusions

We studied a three variable non-linear map which includes a slow, homeostaticmodulation of bistable fast dynamics. This element could be used to representdi!erent behaviors observed in biological cells such as bursts and spikes with plateau.We proposed a synaptic representation that preserves the coupled maps latticecharacter of the network. We are presently applying this formalism to networks withbiological realistic architectures.

Acknowledgements

This research has been supported by FAPESP and CNPq.

References

[1] K. Aihara, T. Takabe, M. Toyoda, Chaotic neural networks, Phys. Lett. A 144 (1990) 333}340.[2] D.R. Chialvo, Generic excitable dynamics on a two-dimensional map, Chaos Solitons Fract. 5 (1995)

461}480.


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[3] J.L. Hindmarsh, R.M. Rose, A model of neuronal bursting using three coupled "rst order di!erentialequations, Proc. Roy. Soc. London B 221 (1984) 87}102.

[4] K. Kaneko, Relevance of dynamic clustering to biological networks, Physica D 75 (1994) 55}73.[5] J. Keener, J. Sneyd, Mathematical Physiology, Springer, New York, 1998.[6] O. Kinouchi, M. Tragtenberg, Modeling neurons by simple maps, Int. J. Bifurcation Chaos 6 (1996)

2343}2360.[7] J. Rinzel, B. Ermentrout, Analysis of neural excitability and oscillations, in: C. Koch, I. Segev (Eds.),

Methods in Neuronal Modeling: From Ions to Networks, MIT Press, Cambridge, 1989.

Silvia M. Kuva was born in Sa� o Carlos, SP, Brazil. She received her M.Sc. degree in Physics in 1993 and herDoctorate degree in Physics in 1998 from the Universidade de Sa� o Paulo. Currently, she is a post-doctoralresearcher at the Department of Physics and Mathematics of the Universidade de Sa� o Paulo at Ribeira� oPreto, Brazil. Her research interests are statistical mechanics models of neural networks and computationalneuroscience.

Gilson F. Lima was born in RondonoH polis, MT, Brazil. He received his M.Sc.degree in science and engineering of materials from the Universidade de Sa� o Pauloin 1997. He is currently working on his Doctorate in the Department of PhysicsandMathematics of the Universidade de Sa� o Paulo at Ribeira� o Preto, performingsimulations of biologically inspired models. He is a teacher in the Escola TeH cnicade CuiabaH , MT, Brazil, since 1993.

Osame Kinouchi was born in Sa� o Paulo, SP, Brazil. He received his M.Sc. degreein Physics in 1992 and his Doctorate degree in Physics in 1996 from the Univer-sidade de Sa� o Paulo. Currently, he is a post-doctoral researcher at the Departmentof Physics and Mathematics of the Universidade de Sa� o Paulo at Ribeira� o Preto,Brazil. His research interests are statistical mechanics applied to learning theory,dynamical systems approach in neural networks and self-organized criticality.

Marcelo H. R. Tragtenberg got his Doctorate degree at the Universidade de Sao Paulo, Brazil. He isa teacher at the Departament of Physics of the Universidade Federal de Santa Catarina, Brazil, andcurrently is a post-doctoral researcher at the Department of Theoretical Physics of Oxford University. Hisresearch interests are neural modeling, magnetic modulated systems, dynamical systems and more recentlysimulation of complex #uids.

Anto( nio C. Roque was born in Sa� o Paulo, SP, Brazil. He received his BS degree in physics from theUniversidade Estadual de Campinas, Brazil, in 1985, and his Ph.D. degree in cognitive science and arti"cialintelligence from the University of Sussex, Brighton, UK, in 1992. He joined the faculty of the Departmentof Physics and Mathematics of the Universidade de Sa� o Paulo at Ribeira� o Preto, Brazil, in June, 1993,where he founded and is the current coordinator of the Laboratory of Neural Networks and Computa-tional Neuroscience. His research interests are computational neuroscience and neural networks applica-tions in medicine.


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A minimal model for excitable and bursting elements